Comprehensive Guide: How to Find Energy in a Quantum Hall Effect

The quantum Hall effect is a remarkable phenomenon in condensed matter physics, where the Hall resistance of a two-dimensional electron system exhibits precise quantization at low temperatures and high magnetic fields. This quantization is related to the energy levels of the system, known as Landau levels, and the filling factor of these levels. By measuring the Hall voltage and the channel current, one can determine the energy of the quantum Hall state and the energy gap between the Landau levels.

Understanding the Quantum Hall Effect Formulas

The key to finding the energy in a quantum Hall effect is to use the following formulas:

1. Hall Resistance Formula:
   ( R_{m xy}={\frac {V_{m text{Hall}}}{I_{m channel}}}={\frac {h}{e^{2}\nu }} )
   Where:
2. ( R_{m xy} ) is the Hall resistance
3. ( V_{m text{Hall}} ) is the Hall voltage
4. ( I_{m channel} ) is the channel current
5. ( e ) is the elementary charge
6. ( h ) is the Planck constant

7. ( \nu ) is the filling factor, which can take on integer or fractional values

8. Energy of the Quantum Hall State:
   ( E_{m n}=-e{\frac {V_{m text{Hall}}}{n}}=-e{\frac {R_{m xy}I_{m channel}}{n}}=-{\frac {h}{n\tau }} )
   Where:

9. ( E_{m n} ) is the energy of the ( n )th quantum Hall state

10. ( \tau ) is the scattering time

11. Energy Gap between Landau Levels:
    ( ΔE=ℏωB={\frac {ℏeB}{m}}={\frac {ℏ}{ml_{m B}^{2}}} )
    Where:

12. ( ΔE ) is the energy gap between the Landau levels
13. ( ωB ) is the cyclotron frequency
14. ( lB ) is the magnetic length

15. ( m ) is the effective mass of the charge carriers

16. Quantum of Flux:
    ( Φ0=2πℏe )
    Where:

17. ( Φ0 ) is the quantum of flux
18. ( e ) is the elementary charge
19. ( ℏ ) is the Planck constant

Measuring the Quantum Hall Effect Parameters

To find the energy in a quantum Hall effect, you need to measure the following parameters:

1. Hall Voltage (( V_{m text{Hall}} )): This can be measured using a voltmeter connected across the Hall contacts of the sample.

2. Channel Current (( I_{m channel} )): This can be measured using an ammeter or a current source connected in series with the sample.

3. Magnetic Field (( B )): The magnetic field can be measured using a gaussmeter or a Hall probe.

4. Effective Mass (( m )): The effective mass of the charge carriers in the material can be determined through other experimental techniques, such as cyclotron resonance or Shubnikov-de Haas oscillations.

5. Scattering Time (( \tau )): The scattering time can be estimated from the width of the Landau level broadening or the mobility of the charge carriers.

Calculating the Energy in the Quantum Hall Effect

Once you have measured the necessary parameters, you can use the formulas provided earlier to calculate the energy in the quantum Hall effect:

1. Hall Resistance (( R_{m xy} )): Use the Hall voltage and channel current to calculate the Hall resistance using the formula:
   ( R_{m xy}={\frac {V_{m text{Hall}}}{I_{m channel}}} )

2. Energy of the Quantum Hall State (( E_{m n} )): Use the Hall resistance, channel current, and the filling factor ( \nu ) to calculate the energy of the ( n )th quantum Hall state:
   ( E_{m n}=-e{\frac {R_{m xy}I_{m channel}}{n}} )

3. Energy Gap between Landau Levels (( ΔE )): Use the magnetic field and the effective mass of the charge carriers to calculate the energy gap between the Landau levels:
   ( ΔE={\frac {ℏeB}{m}} )

4. Quantum of Flux (( Φ0 )): Calculate the quantum of flux using the elementary charge and the Planck constant:
   ( Φ0=2πℏe )

Examples and Numerical Problems

Here are some examples and numerical problems to help you understand the calculations involved in finding the energy in a quantum Hall effect:

Example 1:
Suppose you have measured the following parameters in a quantum Hall effect experiment:
– Hall voltage (( V_{m text{Hall}} )) = 10 mV
– Channel current (( I_{m channel} )) = 1 mA
– Magnetic field (( B )) = 10 T
– Effective mass (( m )) = 0.067 ( m_e ), where ( m_e ) is the mass of an electron

Calculate:
1. The Hall resistance (( R_{m xy} ))
2. The energy of the quantum Hall state (( E_{m n} )) for ( n = 1 )
3. The energy gap between the Landau levels (( ΔE ))
4. The quantum of flux (( Φ0 ))

Example 2:
In a fractional quantum Hall effect experiment, the following parameters were measured:
– Hall voltage (( V_{m text{Hall}} )) = 5 mV
– Channel current (( I_{m channel} )) = 0.5 mA
– Magnetic field (( B )) = 15 T
– Filling factor (( \nu )) = 1/3
– Scattering time (( \tau )) = 1 ps

Calculate:
1. The Hall resistance (( R_{m xy} ))
2. The energy of the quantum Hall state (( E_{m n} )) for ( n = 1 )

Numerical Problem:
A two-dimensional electron system exhibits the quantum Hall effect at a magnetic field of 8 T. The effective mass of the charge carriers is 0.05 ( m_e ), and the scattering time is 0.1 ns. The Hall voltage is measured to be 20 mV, and the channel current is 2 mA.

1. Calculate the Hall resistance (( R_{m xy} )).
2. Determine the energy of the quantum Hall state (( E_{m n} )) for ( n = 1 ).
3. Find the energy gap between the Landau levels (( ΔE )).
4. Calculate the quantum of flux (( Φ0 )).

Remember to show your work and provide the final answers with the appropriate units.

Conclusion

The quantum Hall effect is a fascinating phenomenon that has led to the development of new standards for electrical resistance and has provided insights into the fundamental nature of quantum mechanics. By understanding the formulas and the necessary measurements, you can accurately determine the energy in a quantum Hall effect system. The examples and numerical problems provided in this guide should help you develop a deeper understanding of this topic and apply the concepts in your own research or studies.

References

1. Experimental studies of the fractional quantum Hall effect in the first excited Landau level
2. Quantum Hall Effect
3. Quantum Hall Effect
4. Quantum Hall Effect
5. Quantum Hall Effect